I am going to discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N>>1-dimensional Gaussian landscape.  The particular attention will be paid to the case of landscapes with logarithmically growing correlations and to its recent generalizations.  Those landscapes give rise to a rich multifractal spatial structure of the associated Boltzmann-Gibbs measure.  If time allows, I will also briefly discuss a one-dimensional variant of the model which can be analyzed by a mapping to the Dyson's Coulomb gas.  In particular, in the latter case we conjecture the explicit form of the distribution of the lowest minimum in such a potential.